A Banach algebra is an algebra over the real or complex numbers which is equipped with a complete norm such that |xy| ≤ |x||y|. The study of Banach algebras is a major topic in functional analysis. If you are about to ask a question on C*-algebras or von Neumann algebras please use (c-star-algebras) ...

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146 views

Decomposing $\mathcal{B}(H)$

Let $H$ be an infinite-dimensional Hilbert space and let $\mathcal{B}(H)$ be the (C*/W*-)algebra of bounded operators on it. Actually, you may forget about the involution in $\mathcal{B}(H)$ because I ...
5
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48 views

Finite dimensionality of some subspace of convolution Banach algebra $L^1(G)$

Let $G$ be a locally compact group (not only compact group) with the left Haar measure $\lambda$. Consider the convolution Banach algebra $L^1(G,\lambda)$. For which $f\in L^1(G,\lambda)$ the ...
5
votes
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229 views

Is the left regular representation of an algebra, always faithful?

Let $\mathcal{A}$ be a unital associative algebra with a countable basis $\mathcal{b}$ over $\mathbb{C}$. Let $H=l^2(b)$ be the Hilbert space generated by $\mathcal{b}$. Let $H_0 = \{v \in H \ \vert \ ...
5
votes
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235 views

Invertibility of elements in a Banach algebra

Let $X=L^1\cap L^2$, and $\hat{X}$ be the Banach algebra of the image under Fourier transform of $X$. Then do the unital extension $1\dot{+}\hat{X}$ of $X$ by adding a constant function with the norm ...
4
votes
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22 views

Examples of a Banach space with an algebra structure having only left continuity

There is a theorem (see for example, Rudin's Functional Analysis, theorem 10.2 ) that if $A$ is a Banach space with an algebra structure, such that both left and right multiplication are continuous, ...
4
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162 views

trace norm and tensor product

Let $(M_n (\mathbb{C}), n\|.\|)$ , $(M_n (\mathbb{C}), n\|.\|)$ and $(M_{nm} (\mathbb{C}), nm\|.\|)$ be three Banach algebras. where $$\|A\| = \mathrm{tr}\sqrt{(A^* A)}. $$ What is the norm of $\phi$ ...
4
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37 views

Compactum of Banach algebra

I need an example of Banach algebra $A$ and a left non-trivial closed ideal $I$ with all of following properties: There exists a bounded approximate identity in $I$ for $I$ i.e., a net ...
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55 views

Ideal in Matrix algebra

Let A be a Banach algebra and suppose that $\Lambda$ be a nonempty set. With the following product and norm the matrix space ...
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85 views

On maximal ideal spaces of a banach algebra

I am reading this article on maximal ideal spaces and there is this part that I don't quite understand very well, hope you guys can help me out. "Let $M(A)$ denote the maximal ideal space of a ...
4
votes
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130 views

Is the range of the Fourier transform $L^1(\mathbb{R}) \to C_0(\mathbb{R})$ closed under “quasi-inversion”?

Definitions: A function $f \in C_0(\mathbb{R})$ is quasi-invertible if $1 \notin \operatorname{ran}(f)$. The quasi-inverse of such an $f$ is $\frac{f}{f-1}$. Some discussion: Let $C_1(\mathbb{R})$ ...
4
votes
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101 views

Open map in Banach algebra

I'm having trouble showing a certian function is open and can be extended. Let $\Omega$ be a completely regular topological space and $A=C_b(\Omega)$ the space of all complex-values bounded ...
4
votes
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127 views

Biduals generated by projections

This question is motivated by a similar question recently posed at MO: http://mathoverflow.net/questions/122091/masas-in-second-duals-of-banach-algebras In this setting, let $B$ be a Banach algebra ...
4
votes
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167 views

Does the non-emptiness of the spectrum of an element of a Banach algebra depend on the Axiom of Choice?

One of the most basic results in functional analysis states that the spectrum of any element of a Banach algebra is non-empty. The proof, as most people might have seen, makes use of Liouville's ...
3
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29 views

Unital amenable Banach algebras which is a proper two sided ideal in its second dual

I need some examples of "unital amenable Banach algebras which is a proper two sided ideal in its second dual".
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15 views

Definition of $K_1(A)$ for a Banach algebra $A$

When defining $K_1(A)$ for a Banach algebra $A$, one may consider $\bigcup_{n\in\mathbb{N}}\{x\in GL_n(A^+):x\equiv I_n\mod M_n(A)\}$ and take the quotient by the component containing the identity, or ...
3
votes
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30 views

Derivation into dense ideal of Banach algebras

Let $A$ be a Banach algebra and $I$ be an ideal of $A$. A derivation $D:A\to I$ is a linear bounded map, with the following property: $$D(ab)=aD(b)+D(a)b,\qquad a,b\in A.$$ Suppose that $I$ is dense ...
3
votes
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28 views

Semiprimitivity of second dual of semiprime Banach algebras

Let $A$ be a Banach algebra. Then $A^*$ is right Banach $A$-module with product $\langle b,f.a\rangle=\langle ab,f\rangle$ for every $a,b\in A, f\in A^*$. Define $\langle a,F*f\rangle=\langle ...
3
votes
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96 views

Prove that a norm makes a space Banach

I have to prove that if $A$ is a C*-Algebra then the algebra $A_1$ obtained adjoining the identity is a C*Algebra too (with the usual algebraic operation defined). I have any problem in all the ...
3
votes
0answers
75 views

Do inclusions of Banach algebras preserve spectral radius?

Let $f : A_1 \to A_2$ be an injective homomorphism of unital Banach algebras. It's a standard fact that if $f$ is has closed range, i.e. $A_1$ is embedded as a closed subalgebra of $A_2$, then for ...
3
votes
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196 views

C* algebra of bounded Borel functions

Let $T\in B(H)$ is normal, and $B(\sigma(T))$ denote the $C^*$ algebra of all bounded Borel functions on $\sigma(T)$. Then is it true that $B(\sigma(T))$ is a closed $C^*$ algebra under the sup. norm ...
3
votes
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69 views

In relation with the set of Fredholm perturbation elements

Let $A$ and $B$ be two unital Banach algebras and $T\colon A\to B$ an homomorphism of Banach algebras. Let denote the set of Fredholm perturbation elements in $A$, i.e. $\operatorname{Ft}:=\{r\in ...
3
votes
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155 views

Topology of maximal ideal space

We know that strictly positive integers with multiplication form a semigroup $S$. Let $\mathscr A=\ell ^1(S)$ with convolution. What is $\mathscr A$'s maximal ideal space? It seems enough to find ...
3
votes
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68 views

Let $K$ be a circle. Describe the spectra of two subalgebras of $C(K)$

Suppose $K=\{\lambda\in\mathbb{C}: 1<\vert\lambda\vert<2\};$ put $f(\lambda)=\lambda$. Let $A$ be the smallest closed subalgebra of $C(K$) that contains $1$ and $f$. Let $B$ be the smallest ...
3
votes
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110 views

Density of operators

I am interested in operators on non-reflexive Banach space. Let $X$ be a Banach space and let $L(X)$ be the algebra of operators acting on $X$. We may embed $L(X)$ into $L(X^{**})$ by ...
2
votes
0answers
16 views

A condition on quotient norms on quotient Banach algebras

Let $A$ be a non-unital Banach algebra, and let $A^+$ be the unitization of $A$ consisting of elements of the form $(a,z)$ where $a\in A$ and $z\in\mathbb{C}$ with multiplication ...
2
votes
0answers
24 views

Path of completely bounded maps has uniformly bounded cb norm?

If $\phi_t:A\rightarrow B$ is completely bounded for $t\in[0,1]$, and $t\mapsto\phi_t(a)$ is continuous for each $a\in A$, is $\sup_{t\in[0,1]}||\phi_t||_{cb}$ finite? Here, $A$ and $B$ are operator ...
2
votes
0answers
28 views

What is the closure of the absolutely convex hull of this set?

Let $E$ be a Banach algebra, and $e_0\in E$, $e_{0}\neq0$. Suppose $0<r<\|e_{0}\|\leq 1$. Let $E_{1}=\{e\in E:\|e\|=1\}$. Clearly, $e_{0}\notin rE_{1}$. Consider the set ...
2
votes
0answers
21 views

Operator norm of matrix of scalars regarded as matrix with entries in the unitization of a Banach algebra

Let $A$ be a non-unital Banach algebra, and let $A^+=\{(a,z):a\in A,z\in\mathbb{C}\}$ with product $(a,z)(b,w)=(ab+zb+wa,za)$ and norm $||(a,z)||=||a||+|z|$. Equip $M_n(A^+)$ with the operator norm by ...
2
votes
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13 views

Approximation by elements in intersection of two Banach subalgebras

Let $A$ be a Banach algebra, and let $A_1,A_2$ be Banach subalgebras of $A$. Suppose that there exists $c>0$ such that whenever $a_i\in A_i$ ($i=1,2$) and $||a_1-a_2||<\varepsilon$, then there ...
2
votes
0answers
24 views

Norm of homomorphism embedding Banach algebra into matrix algebra

Let $A$ be a nonunital Banach algebra, and let $\tilde{A}$ be the unitization of $A$, i.e. $\tilde{A}=\{(a,\lambda):a\in A,\lambda\in\mathbb{C}\}$ with multiplication given by ...
2
votes
0answers
32 views

For which $F$ we have $F(f\ast g)= f\ast F(g)$ for all $f,g \in L^{1}$?

Young's inequality tells us that: $L^{1}\ast L^{p} \subset L^{p}, (1\leq p \leq \infty)$ My Question: What are examples of functions $F:L^{1}(\mathbb R)\to L^{p}(\mathbb R)$ with the property ...
2
votes
0answers
42 views

Clarification on Wolfram Mathworld's explanation of the connection between Gelfand Transform and Fourier Transform

http://mathworld.wolfram.com/GelfandTransform.html In the definition, what does $x$, $\hat x(\phi)$, and $\phi$ represent exactly if we were to consider definition of the Fourier transform? Can ...
2
votes
0answers
27 views

equivalence of properties. Is the restriction in (ii) redundant?

I have a question about the claim, which I found in a paper: Let $\phi:A\to B$ a linear map between $C^*$-algebras $A$ and $B$. The following are equivalent: ...
2
votes
0answers
42 views

Maximal ideal space of $L^{\infty}(m)$ separable?

Let $m$ denote the Lebesgue-measure on the unit interval and let $L^{\infty}(m)$ be the set of measurable essentially bounded functions on that interval (i.e. $f \in L^{\infty}(m)$ if and only if ...
2
votes
0answers
55 views

Prove that disk algebra is isomorphic to the closure of $\mathbb{C}(z)$ in $C(\mathbb{T})$.

Let $D = \{ z \in \mathbb{C} : |z| < 1 \}$ be the open unit disk and $\mathbb{T} = \{ z \in \mathbb{C} : |z| = 1 \}$ its boundary. We will naturally write $\bar{D}$ for its closure $\{ z \in ...
2
votes
0answers
38 views

Proof of theorems in the field of banach-and $c^*$-algebras in a categorial language

At the moment I'm studying the basics in the theory of banach- and $c^*$-algebras. There are many results in the theory of $c^*$-algebra which you first prove in the unital case and then in the ...
2
votes
0answers
39 views

positive elements in $L(H)$

At the moment I'm studying positive elements in $C^*$-algebras. Let $H$ be a complex Hilbert space, $L(H)$ the linear bounded operators $H\to H$ and $T^*=T$. Then: $$\sigma(T)\subseteq [0,\|T\| ]\; ...
2
votes
0answers
22 views

When do injective and projective tensor norms agree?

For $C^*$-algebra tensor products, one talks about the min and max tensor norms, and they agree when one of the $C^*$-algebras is nuclear. For general Banach algebras, what is the analog of ...
2
votes
0answers
39 views

Banach algebras for which the Gelfand transform is 1-1 but not onto

Are there examples of Banach Algebra's for which the Gelfand transform is 1-1, (ie: the intersection of all maximal ideals of the algebra is $\{0\}$) but not onto? Context Page 96 of Kaniuth's ...
2
votes
0answers
52 views

Subalgebras of $C_b(X)$ whose elements do not vanish simultaneously at any point

Let $X$ be a completely regular space. How can I find all Banach subalgebras of $C_b(X)$ (all complex-valued bounded continuous functions on $X$) with the property that for every $x\in X$ there ...
2
votes
0answers
162 views

Semigroups: Entire Elements (I)

Problem Given a Banach space $E$. Consider a C0-semigroup: $$T:\mathbb{R}_+\to\mathcal{B}(E)$$ Define its generator by: $$Ax:=\lim_{h\downarrow0}\frac{1}{h}(T(h)x-x)\in E$$ (It is a densely-defined ...
2
votes
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105 views

Riesz Projection in Functional Analysis.

By definition the Riesz projection of a Banach algebra element $a$ associated with a complex number $\alpha$ is given by $p(\alpha , a)= \frac{1}{2\pi i} \int_{\Gamma} ( \mu -a)^{-1} \,\, ...
2
votes
0answers
80 views

Using Stone–Weierstrass theorem for completely regular space

Let $X$ be completely regular. If $K$ is a compact subset of $X$, define $$p_K(f)=\sup\{|f(x)|;x\in X\}$$ then $\{p_k; \text{K is a compact }\}$ is a family of seminorms that makes $C(X)$ into a ...
2
votes
0answers
40 views

How I can make $\Bbb{C}[x]$ into a Banach algebra?

Let $\Bbb{C}$ the complex field. Define $\Bbb{C}[x]$ as the set of all polynomials with variable $x$. It is known that $\Bbb{C}[x]$ is a algebra. Now the question is this that how I can make it a ...
2
votes
0answers
25 views

How to prove the set of fourier multipliers is a banach algebra?

Hi I am new here at math stack Exchange, this is my first question, hope you guys can help me out:) Suppose $F\colon L^2(\mathbb{R} ) \to L^2(\mathbb{R})$ is the Fourier transform given by ...
2
votes
0answers
32 views

Can we expect, $\left\|f|f|- g|g|\right\|\leq C \left \|f-g\right \|$ in the Banach algebra $A(\mathbb R)$?

Let $f\in L^{1}(\mathbb R)$ and it Fourier transform, $\hat{f} (y) : = \int _ {\mathbb R} f(x) e^{-2\pi i x\cdot y} dx ; y \in \mathbb R ;$ and consider Fourier algebra $$A(\mathbb R):= \{f\in ...
2
votes
0answers
88 views

Find an example of product of operators not Jointly Continuous in strong topology.

I'm trying to find an example of the fact that the product of operators is not jointly continuous in the strong topology. I know the example of the unilateral shift (that is on wikipedia), but I ...
2
votes
0answers
72 views

How to prove $\widehat{(fg)}(n) = \hat{f} (n) \ast \hat{g} (n); (n \in \mathbb Z)$?

Let $f, g \in L^{1}(\mathbb T)= L^{1} ([-\pi, \pi))$. We define, the Fourier transform of $f$ as follows: $$\hat{f}(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi} f(t) e^{-int} dt, \ (n\in \mathbb Z).$$ It is ...
2
votes
0answers
173 views

Functional analysis problem involving maximal ideal space

For the past week, I've been trying to solve, as a practice homework exercise, Problem 6 of Chapter 11 in Rudin's Functional Analysis, but have not gotten very far it seems. The problem is as follows: ...
2
votes
0answers
79 views

Banach algebra.

Iam new in this field. I am reading a paper and have encoutered the following Lemma. Let $u\in F_{1}.$ Then $Sp(u)=\{0, tr(u)\},$ where $F_{1}$ is the set of one-dimensional elements and tr(u) is the ...